Existence of Two Solutions for a Critical Elliptic Problem with Nonlocal Term in
In this paper, we prove the existence of two positive solutions for a critical elliptic problem with nonlocal term and Sobolev exponent in dimension four.
In this work, we are mainly concerned by the existence and the multiplicity of solutions for the following critical elliptic nonlocal problem: where is a smooth bounded domain of , and are positive constants, and belongs to satisfying suitable condition specified afterward.
In our setting, the Laplacian operator is associated to Kirchhoff term , which contains an integral over the entire domain this implies that the equation in is no longer a pointwise identity and so the problem turns to be nonlocal. This fact brings some mathematical difficulties in the search of the solution, and the solvability of this kind of problems has been under various authors’ attention; so, some classical investigations can be seen in the works [1, 2] and the references therein.
Such nonlinear Kirchhoff’s equations can be used for describing the dynamic for an axially moving string and was first formulated by Kirchhoff himself  in 1883, he take into account the changes in length of the strings produced by transverse vibrations, and his model can be seen as a generalization of the classical D’Alembert wave equation for free vibrations of elastic strings.
Problems which involve nonlocal operator have been widely studied due to their numerous and relevant applications in various fields of sciences. In particular, Kirchhoff type problems proved to be valuable tools for modeling several physical and biological phenomena, and many works have been made to ensure the existence of solutions for such problems; we quote in particular the article of Lions . Since this famous paper, very fruitful development has given rise to many works on this advantageous axis, and in most of them, the used approach relies on topological methods. However, just few improvements were held concerning the multiplicity of solutions. In , Maia obtained a multiplicity of solutions for a class of -Choquard equations with a nonlocal and nondegenerate Kirchhoff term by using truncation arguments and Krasnoselskii’s genus. In , Vetro studied the existence of two different notions of solutions by using Galerkin approximation method, jointly with the theory of pseudomonotone operators.
With this regard, variational approach was solicited instead of topological methods to solve this kind of problems and also to prove the existence of multiple solutions; we refer interested readers to the works [7–10].
We begin by giving an overview about the previous research related to the problem which can be written in the more general form,
where is a smooth bounded domain of
Without nonlocal term , much interest has grown on problems involving critical exponents, and there are many publications dealing with the existence of solutions, starting from the celebrated paper by Brézis and Nirenberg  when and are the critical Sobolev exponent. For convenience of the reader, we give a brief summary of these results: they established existence results in dimension when is a ball namely, and they ensure the existence of a positive constant such that the problem admits a positive solution for where is the first eigenvalue of the operator In higher dimensions they proved the existence of a positive solution for sufficiently small, i.e., and no positive solution for and a star-shaped domain.
When Ambrosetti et al.  established a multiplicity result in a bounded domain of indeed, they ensured the existence of a positive constant such that the problem admits two positive solutions for a positive solution for and no positive solution for
For the nonhomogeneous case, namely, when Tarantello  proved the existence of at least two solutions when satisfies
We emphases that the extension of the previous results to the nonlocal case, namely, for elliptic problems driven by Kirchhoff type operator are not obvious in high dimensions Therefore, no improvement was hold concerning the multiplicity of solutions in this case.
For the case and , Naimen in  treated the problem for and obtained homologous results than the ones obtained by Brézis and Niremberg  in the nonlocal case under a suitable condition on .
In dimension four, Naimen in  used variational methods to explore problem and showed that admits a positive solution when
In the same order of ideas and still in the nonlocal case Lei et al. in  and Liao et al. in  extended the findings of  to a more general setting, namely, with the Kirchhoff operator; they established a multiplicity result in dimensions three and four, respectively.
Benmansour and Bouchekif  generalized the results obtained by Tarantello  to the nonlocal case in dimension three. Indeed, they have shown the existence of two solutions under a sufficient condition on by introducing the Nehari manifold.
A natural question is to know whether the multiplicity result persists in the case of dimension four.
In the current paper, our main purpose inspired by  is to see that the result obtained in  can be extended to dimension four. We emphases that our results are new and complement the above works.
In order to study we shall work with the space endowed with the norm we use also the following notation: for , and denote generic positive constants whose exact values are not important, is the ball of center and radius , denotes any quantity which tends to zero as tends to infinity, and is the best Sobolev constant defined by
To state the main results, we define where , a small enough positive number and belongs to
The main results are concluded as the following theorems, which are news for the case when
Theorem 1. Assume that . Then, the problem admits a local minimal solution with Furthermore for
Theorem 2. Assume that . Then, the problem admits another solution with Furthermore, for
Notice that if
Moreover, the assumption certainly holds if satisfies certain conditions, for example,
for all with Indeed, we have is achieved and strictly positive if satisfies (see Lemma 2.2 in ). In order as then,
This paper is structured as follows: in Section 2, we give some basic results useful for what follows. Section 3 is devoted to the proofs of our main results.
2. Some Preliminary Results
We consider the energy functional associated to problem defined for and given by
Observe that , whose derivative at the point is given by
Obviously, if is a critical point of the functional ; then, is a weak solution of problem
In general, is not bounded from below on , to overcome this and achieve a multiplicity result, the key argument is to use an appropriate manifold called in mathematical literature the Nehari manifold, it is a suitable manifold who has a pertinent property to prove the distinction of two solutions. Indeed, a minimizer in this set may give rise to solution of the corresponding equation. This so called Nehari manifold is defined by
Lemma 3. Assume that , and . Then, the functional is coercive and bounded from below on
Proof. For , we have
Thus, is coercive and bounded from below on .
Let for and These maps are known as fibering maps and were first introduced by Drábek and Pohozaev  The set is closely linked to the behavior of , for more details, see for example  or .
It is natural to split into three subsets: where These subsets correspond to local minima, points of inflexion, and local maxima of , respectively.
Definition 4. A sequence is said to be a Palais Smale sequence at level ( sequence in short) for if verifies Palais Smale condition at level ( condition in short) if any sequence has a convergent subsequence in
Next, for , and , a small enough positive number set then Easy computations show that is concave and achieves its maximum at the point where
Now, for set , that is
Fix , then, for , we have so
This is crucial for the following.
Lemma 5. Assume that , then,
Proof. Arguing by contradiction we assume that there exists i.e., verifies From (24) and (25), we derive that As , we get from (24), (26), and the definition of that is Thus, as and , we derive that therefore, from (25), we obtain with Then, from (23), (25), and (26), we get which yields to a contradiction.
Lemma 6. Assume that , then, for all there exists unique positive value such that Moreover, if then, there exists unique positive value such that
Proof. We have and is concave and achieves its maximum at the point If ; then, there exists a unique such that and , which implies that and for all Moreover, if then, there exists a unique such that and , which implies that and for all
Set In the following lemma, we prove that is closed and disconnects in exactly two connected components and .
Lemma 7. Assume that . Then (i) is closed(ii)(iii)
Proof. Let and then, . Assume by contradiction that then
So, this implies that
From (36) and the definition of , we get so which yields to a contradiction.
Let and then, , and there exists unique such that As then Thus if and, then, and Let then Since it follows that So, , and we conclude that
By Lemma 6, we know that and are not empty, so we can define with
Lemma 8. Assume that , then, there exists such that
Proof. Let then
Set the unique solution of the equation it follows By Lemma 6, there exists a unique positive value such that . So consequently
The following lemma is needed for prove the existence of Palais Smale sequences.
Lemma 9. Assume that . Then, for any there exist and a differentiable function such that
Proof. Let and define as follows Clearly, Moreover, from Lemma 5, we derive that Thus, we get our result by a straightforward application of the implicit function theorem to the function at the point .
Lemma 10. Let There exist a Palais Smale sequences such that
Proof. Assume by Lemma 3, is bounded from below in then by applying the Ekeland Variational Principle, we can obtain a minimizing sequence satisfying
for all Thus,
By using Lemma 8, we get for large enough this implies that then and by Holder inequality, we get Consequently, and Now, we show that tend to as goes to . Arguing by contradiction and fix with .
Then, by Lemma 9, there exist and a function such that with and . By (56) and the Taylor expansion of , we have Then We have This, together with (61) implies for a suitable constant Now, we must show that is uniformly bounded in indeed, since is a bounded sequence, we have from Lemma 8for a suitable constant Assume by contradiction that for a subsequence still called we have Then, as is a small enough positive number, we get So from (61), we derive that Also, as , we get from (68) then which is absurd. At this point, we conclude that in
For adopting exactly the same way as in the case where .
In the following, we will prove our results.
3. Proofs of the Main Results
The proof of our main results is divided in two parts.
3.1. Existence of a Solution in
In this subsection, we prove that has a solution in
Proposition 11. Assume that . Then, the minimization problem attaints its infimum at a point . Moreover, is a local minimizer for in
Proof. By using Lemma 10, there exists a bounded minimizing sequence such that and in So, we deduce that is bounded in
Passing to a subsequence if necessary, we have weakly in , then, for all In addition, from (60), we get . So, is a weak solution for and
Thus then It follows that converges strongly to in , then, and necessarily To conclude that is a local minimum of , let us recall that we have from Lemma 6Choose sufficiently small to have and satisfying for every Since as we can always assume that so and for , we have from (75), we can take and conclude that for all such that Thus, is a local minimum of
If , we have and clearly and from (75) necessarily . Therefore, as , we get so, we can always take
3.2. Existence of a Solution in
The following part is devoted to prove the existence of a second solution such that First, we determine the good level for covering the Palais Smale condition.
We have the following important result.
Lemma 12. Assume that . Then, satisfies the condition for with
Proof. Let be a sequence with then
Hence, is a bounded sequence in Thus for a subsequence still denoted and we can find such that in and a.e in . Therefore, , and
Let From Brézis-Lieb Lemma , one has this implies that Assume that with , then, by (85) and the Sobolev inequality, we obtain this implies that Hence On the other hand, we have consequently, we obtain which is a contradiction. Therefore, and strongly in
Now, it is natural to show that As it is well know, , the best Sobolev constant defined above is attained in by For let such that for for and Now, we shall give some useful estimates of the extremal functions Let and The following estimates are obtained in  as tends to Let a set of positive measure such that on (if not replace and by and , respectively).
Lemma 13. Let a small enough positive number. Assume that satisfies ; then, for every and a.e. there exists such that for every
Proof. For small enough, let us consider the functional defined by
We have by (92)
On the other hand, since is a solution of problem , we have , and
Then we obtain by (92) where